Answer:
1) a = -⅙, b = ⅙, c = 1
2) 6 units
Step-by-step explanation:
1) f(x) = ax² + bx + c
Given the roots, we can write this as:
f(x) = a (x + 2) (x − 3)
We know that f(13) = -25, so we can plug this in to find a:
-25 = a (13 + 2) (13 − 3)
-25 = 150a
a = -⅙
Therefore, the factored form is:
f(x) = -⅙ (x + 2) (x − 3)
Distributing:
f(x) = -⅙ (x² − x − 6)
f(x) = -⅙ x² + ⅙ x + 1
Graph: desmos.com/calculator/6m6tjoodvb
2) Volume of a right prism is area of the base times the height.
V = Ah
The base is an equilateral triangle. Area of a triangle is one half the base times height.
V = ½ ab h
The height of the triangle is the same as the height of the prism.
V = ½ bh²
In an equilateral triangle, the height is equal to half the base times the square root of 3.
V = ½ b (½√3 b)²
V = ⅜ b³
Given that V = 81, solve for b.
81 = ⅜ b³
216 = b³
b = 6
